Xenharmonic Wiki

22edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
SupahstarSaga (talk | contribs)
autopatrolled, emailconfirmed
131 edits
m fixed "orwell diatonic 5L4s 323232322"
TallKite (talk | contribs)
autopatrolled, Sysops
4,223 edits
→Chord Names: added a link to the 22edo chord names page, also did lots of cleanup
Line 96: Line 96:
== Notation ==
== Notation ==


=== Superpyth/Porcupine Notation ===
{| class="wikitable center-all right-2"
The intervals of 22 EDO may be thought of as a system arising from both Superpyth and Porcupine temperament therefore, it makes sense to categorize each on as major and minor of each temperament. s indicates superpyth, p indicates Porcupine, because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.
 
{| class="wikitable center-1 right-3"
|-
|-
! Degree
! Degree
! Name and Abbreviation
! Cents
! Cents
! Approximate Ratios*
! Approximate Ratios*
! colspan="3" |[[Ups and Downs Notation]]
|-
|-
| 0
| 0
| Natural Unison, 1
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
|perfect unison
|P1
|D
|-
|-
| 1
| 1
| s-minor second, sm2
| 54.545
| 54.545
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]
|minor 2nd
|m2
|Eb
|-
|-
| 2
| 2
| p-diminished second, pd2
| 109.091
| 109.091
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]
|upminor 2nd
|^m2
|^Eb
|-
|-
| 3
| 3
| p-minor second, pm2
| 163.636
| 163.636
| [[12/11]], [[11/10]], [[10/9]]
| [[12/11]], [[11/10]], [[10/9]]
|downmajor 2nd
|vM2
|vE
|-
|-
| 4
| 4
| (s/p) Major second, M2
| 218.182
| 218.182
| [[9/8]], [[17/15]], [[8/7]]
| [[9/8]], [[17/15]], [[8/7]]
|major 2nd
|M2
|E
|-
|-
| 5
| 5
| s-minor third, sm3
| 272.737
| 272.737
| [[20/17]], [[7/6]]
| [[20/17]], [[7/6]]
|minor 3rd
|m3
|F
|-
|-
| 6
| 6
| p-minor third, pm3
| 327.273
| 327.273
| [[6/5]], [[17/14]], [[11/9]]
| [[6/5]], [[17/14]], [[11/9]]
|upminor 3rd
|^m3
|^F
|-
|-
| 7
| 7
| p-Major third, pM3
| 381.818
| 381.818
| [[5/4]], [[96/77]]
| [[5/4]], [[96/77]]
|downmajor 3rd
|vM3
|vF#
|-
|-
| 8
| 8
| s-Major third, sM3
| 436.364
| 436.364
| [[14/11]], [[9/7]], [[22/17]]
| [[14/11]], [[9/7]], [[22/17]]
|major 3rd
|M3
|F#
|-
|-
| 9
| 9
| Natural Fourth, 4, N4
| 490.909
| 490.909
| [[4/3]]
| [[4/3]]
|perfect fourth
|P4
|G
|-
|-
| 10
| 10
| p-Major Fourth, pM4, s-dim fifth
| 545.455
| 545.455
| [[15/11]], [[11/8]]
| [[15/11]], [[11/8]]
|up-4th, dim 5th
|^4, d5
|^G, Ab
|-
|-
| 11
| 11
| Augmented Fourth, A4, Half-Octave, HO
| 600.000
| 600.000
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]
|downaug 4th, updim 5th
|vA4, ^d5
|vG#, ^Ab
|-
|-
| 12
| 12
| p-minor Fifth, pm5, s-aug fourth
| 654.545
| 654.545
| [[16/11]], [[22/15]]
| [[16/11]], [[22/15]]
|aug 4th, down-5th
|A4, v5
|G#, vA
|-
|-
| 13
| 13
| Natural Fifth, 5, N5
| 709.091
| 709.091
| [[3/2]]
| [[3/2]]
|perfect 5th
|P5
|A
|-
|-
| 14
| 14
| s-minor sixth, sm6
| 763.637
| 763.637
| [[17/11]], [[14/9]], [[11/7]]
| [[17/11]], [[14/9]], [[11/7]]
|minor 6th
|m6
|Bb
|-
|-
| 15
| 15
| p-minor sixth, pm6
| 818.182
| 818.182
| [[8/5]], [[77/48]]
| [[8/5]], [[77/48]]
|upminor 6th
|^m6
|^Bb
|-
|-
| 16
| 16
| p-Major sixth, pM6
| 872.727
| 872.727
| [[18/11]], [[28/17]], [[5/3]]
| [[18/11]], [[28/17]], [[5/3]]
|downmajor 6th
|vM6
|vB
|-
|-
| 17
| 17
| s-Major sixth, sM6
| 927.273
| 927.273
| [[17/10]], [[12/7]]
| [[17/10]], [[12/7]]
|major 6th
|M6
|B
|-
|-
| 18
| 18
| (s/p) minor seventh, m7
| 981.818
| 981.818
| [[7/4]], [[30/17]], [[16/9]]
| [[7/4]], [[30/17]], [[16/9]]
|minor 7th
|m7
|C
|-
|-
| 19
| 19
| p-Major seventh, pM7
| 1036.364
| 1036.364
| [[9/5]], [[11/6]], [[20/11]]
| [[9/5]], [[11/6]], [[20/11]]
|upminor 7th
|^m7
|^C
|-
|-
| 20
| 20
| p-Augmented Seventh
| 1090.909
| 1090.909
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]
|downmajor 7th
|vM7
|vC#
|-
|-
| 21
| 21
| s-Major Seventh, sM7
| 1145.455
| 1145.455
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]
|major 7th
|M7
|C#
|-
|-
| 22
| 22
| Octave, 8
| 1200.000
| 1200.000
| [[2/1]]
| [[2/1]]
|perfect octave
|P8
|D
|}
|}


<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.
<nowiki>*</nowiki> some simpler ratios, ordered by increasing size, based on treating 22-edo as a 2.3.5.7.11.17 subgroup temperament; other approaches are possible.


=== Ups and Downs, Porcupine and Pentatonic Notations ===
=== Superpyth/Porcupine Notation, Porcupine Notation and Pentatonic Notation ===
22edo intervals can also be notated using [[Ups_and_Downs_Notation|ups and downs]]. This notation allows for easy chord naming. The keyboard runs D * * * E F * * * G * * * A * * * B C * * * D. The natural notes represent the conventional chain of 5ths FCGDAEB.
Superpyth/Porcupine Notation is a system arising from both Superpyth and Porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates Porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.


Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D. The natural notes represent a chain of 2nds ABCDEFG.
Another possible notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. This is the only way to use a heptatonic notation without additional accidentals. The keyboard runs D * * E * * F * * G * * * A * * B * * C * * D. The natural notes represent a chain of 2nds ABCDEFG.
Line 235: Line 278:
! [[Degree]]
! [[Degree]]
! [[cent|Cents]]
! [[cent|Cents]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | Superpyth/Porcupine Notation
! colspan="3" | Porcupine
! colspan="3" | Porcupine
! colspan="3" | Pentatonic
! colspan="3" | Pentatonic
Line 241: Line 284:
| 0
| 0
| 0
| 0
| perfect unison
| Natural Unison
| P1
| 1
| D
| perfect unison
| perfect unison
| P1
| P1
Line 253: Line 295:
| 1
| 1
| 55
| 55
| minor 2nd
| s-minor second
| m2
| sm2
| Eb
| aug unison
| aug unison
| A1
| A1
Line 265: Line 306:
| 2
| 2
| 109
| 109
| upminor 2nd
| p-diminished second
| ^m2
| pd2
| ^Eb
| dim 2nd
| dim 2nd
| d2
| d2
Line 277: Line 317:
| 3
| 3
| 164
| 164
| downmajor 2nd
| p-minor second
| vM2
| pm2
| vE
| perfect 2nd
| perfect 2nd
| P2
| P2
Line 289: Line 328:
| 4
| 4
| 218
| 218
| major 2nd
| (s/p) Major second
| M2
| M2
| E
| aug 2nd
| aug 2nd
| A2
| A2
Line 301: Line 339:
| 5
| 5
| 273
| 273
| minor 3rd
| s-minor third
| m3
| sm3
| F
| dim 3rd
| dim 3rd
| d3
| d3
Line 313: Line 350:
| 6
| 6
| 327
| 327
| upminor 3rd
| p-minor third
| ^m3
| pm3
| ^F
| minor 3rd
| minor 3rd
| m3
| m3
Line 325: Line 361:
| 7
| 7
| 382
| 382
| downmajor 3rd
| p-Major third
| vM3
| pM3
| vF#
| major 3rd
| major 3rd
| M3
| M3
Line 337: Line 372:
| 8
| 8
| 436
| 436
| major 3rd
| s-Major third
| M3
| sM3
| F#
| aug 3rd, dim 4th
| aug 3rd, dim 4th
| A3, d4
| A3, d4
Line 349: Line 383:
| 9
| 9
| 491
| 491
| perfect fourth
| Natural Fourth
| P4
| 4, N4
| G
| minor 4th
| minor 4th
| m4
| m4
Line 361: Line 394:
| 10
| 10
| 545
| 545
| up-4th, dim 5th
| p-Major Fourth, s-dim fifth
| ^4, d5
| pM4, sd5
| ^G, Ab
| major 4th
| major 4th
| M4
| M4
Line 373: Line 405:
| 11
| 11
| 600
| 600
| downaug 4th, <br>updim 5th
| Augmented Fourth,  
| vA4, ^d5
Half-Octave
| vG#, <br>^Ab
| A4, HO
| aug 4th, <br>dim 5th
| aug 4th, <br>dim 5th
| A4, d5
| A4, d5
Line 385: Line 417:
| 12
| 12
| 655
| 655
| aug 4th, down-5th
| p-minor Fifth, s-aug Fourth
| A4, v5
| pm5, sA4
| G#, vA
| minor 5th
| minor 5th
| m5
| m5
Line 397: Line 428:
| 13
| 13
| 709
| 709
| perfect 5th
| Natural Fifth
| P5
| 5, N5
| A
| major 5th
| major 5th
| M5
| M5
Line 409: Line 439:
| 14
| 14
| 764
| 764
| minor 6th
| s-minor sixth
| m6
| sm6
| Bb
| aug 5th, dim 6th
| aug 5th, dim 6th
| A5, d6
| A5, d6
Line 421: Line 450:
| 15
| 15
| 818
| 818
| upminor 6th
| p-minor sixth
| ^m6
| pm6
| ^Bb
| minor 6th
| minor 6th
| m6
| m6
Line 433: Line 461:
| 16
| 16
| 873
| 873
| downmajor 6th
| p-Major sixth
| vM6
| pM6
| vB
| major 6th
| major 6th
| M6
| M6
Line 445: Line 472:
| 17
| 17
| 927
| 927
| major 6th
| s-Major sixth
| M6
| sM6
| B
| aug 6th
| aug 6th
| A6
| A6
Line 457: Line 483:
| 18
| 18
| 982
| 982
| minor 7th
| (s/p) minor seventh
| m7
| m7
| C
| dim 7th
| dim 7th
| d7
| d7
Line 469: Line 494:
| 19
| 19
| 1036
| 1036
| upminor 7th
| p-Major seventh
| ^m7
| pM7
| ^C
| perfect 7th
| perfect 7th
| P7
| P7
Line 481: Line 505:
| 20
| 20
| 1091
| 1091
| downmajor 7th
| p-Augmented Seventh
| vM7
| pA7
| vC#
| aug 7th
| aug 7th
| A7
| A7
Line 493: Line 516:
| 21
| 21
| 1145
| 1145
| major 7th
| s-Major Seventh
| M7
| sM7
| C#
| dim 8ve
| dim 8ve
| d8
| d8
Line 505: Line 527:
| 22
| 22
| 1200
| 1200
| perfect octave
| Octave
| P8
| 8
| D
| perfect octave
| perfect octave
| P8
| P8
Line 515: Line 536:
| D
| D
|}
|}
=== Decatonic Notation ===
The decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.
Chain 1: C G D A E
Chain 2: γ δ α ε β
The alphabet is, in ascending order: C δ D ε E γ G α A β C
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.
==Chord Names==
See also [[22 EDO Chords]], [[Chords of orwell]].


Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Line 521: Line 557:
|-
|-
! quality
! quality
! [[color name]]
![[color name]]
! monzo format
! [[monzo]] format
! examples
! examples
|-
|-
| minor
| rowspan="2" | minor
| zo
| zo
| {a, b, 0, 1}
| [a b 0 1>
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| "
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| [a b> where b &lt; -1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| [a b -1>
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| downmajor
| downmajor
| yo
| yo
| {a, b, 1}
| [a b 1>
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| major
| rowspan="2" | major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| [a b> where b &gt; 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| "
| ru
| ru
| {a, b, 0, -1}
| [a b 0 -1>
| 9/7, 12/7
| 9/7, 12/7
|}
|}


=== Decatonic Notation ===
All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo and ru triads:
The decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.
 
Chain 1: C G D A E
 
Chain 2: γ δ α ε β
 
The alphabet is, in ascending order: C δ D ε E γ G α A β C
 
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G-D is a fifth, and so is γ-δ.
 
==Chord Names==
 
See also [[22 EDO Chords]], [[Chords of orwell]].
 
All 22edo chords can be named using ups and downs notation. Here are the zo, gu, yo and ru triads:


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 610: Line 629:
| C major or C
| C major or C
|}
|}
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).
0-8-13-18 = C E G Bb = C7 = "C seven"
0-7-13-18 = C vE G Bb = Cv,7 = "C down add-seven"
0-8-13-21 = C E G B = CM7 = "C major seven"
0-7-13-20 = C vE G vB = CvM7 = "C downmajor seven"
0-3-13 = C vD G = Cv2 or C(v2) = "C down-two"


0-4-13 = C D G = C2
0-4-13 = C D G = C2
Line 629: Line 636:
0-10-13 = C ^F G = C^4 or C(^4)
0-10-13 = C ^F G = C^4 or C(^4)


0-5-10 = C Eb Gb = Cdim
0-5-10 = C Eb Gb = Cd = Cdim


0-5-11 = C Eb ^Gb = Cdim(^5)
0-5-11 = C Eb ^Gb = Cd(^5)


0-5-12 = C Eb vG = Cm(v5)
0-5-12 = C Eb vG = Cm(v5)


0-5-10-15 = C Eb Gb Bbb = Cdim7
For a more complete list, see [[22edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].
 
0-5-11-14 = C Eb ^Gb vBbb = Cdim7(^5,v7)
 
0-6-11-15 = C ^Eb ^Gb Bbb = Cdim7(^3,^5)
 
0-6-11-16 = C ^Eb ^Gb ^Bbb = C^dim7(^5)
 
0-5-13-17 = C Eb G A = Cm6
 
0-8-13-17 = C E G A = C6
 
0-8-13-16 = C E G vA = C,v6 = C add down-six
 
0-7-13-17 = C vE G A = C6(v3)
 
0-7-13-16 = C vE G vA = Cv6
 
0-5-13-18 = C Eb G Bb = Cm7
 
0-6-13-19 = C ^Eb G ^Bb = C^m7
 
0-5-13-16 = C Eb G vA = Cm,v6 = "C minor add down-6"
 
0-8-13-19 = C E G ^Bb = C,^7 = "C add up-seven"
 
Sometimes doubled ups/downs are unavoidable:
 
0-6-12-15 = C ^Eb vG vvA = Cm6(^3,v5,vv6), or C ^Eb ^^Gb Bbb = Cdim7(^3,^^5)
 
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].


== Just approximation ==
== Just approximation ==
Retrieved from "https://en.xen.wiki/w/22edo"